Operator Algebra seminar

Speaker: Piotr M. Hajac (IMPAN)

Title:There and back again: from the Borsuk-Ulam theorem to quantum spaces

Abstract: Assuming that both temperature and pressure are continuous functions, we can conclude that there are always two antipodal points on Earth with exactly the same pressure and temperature. This is the two-dimensional version of the celebrated Borsuk-Ulam Theorem which states that for any continuous map from the n-dimensional sphere to n-dimensional real Euclidean space there is always a pair of antipodal points on the sphere that are identified by the map. Our quest to unravel topological mysteries in the Middle Earth of quantum spaces will begin with gentle preparations in the Shire of elementary topology. Then, after riding swiftly through the Rohan of C*-algebras and Gelfand-Naimark Theorems, and carefully avoiding the Mordor of incomprehensible technicalities, we shall arrive in the Gondor of compact quantum groups acting freely on unital C*-algebras. It is therein that the noncommutative Borsuk-Ulam-type conjecture dwells waiting to be proven or disproven. After revealing how to prove the conjecture assuming some torsion or local-triviality properties, we shall explain how the general case (no torsion or local-triviality assumptions) implies the famous and long-standing weak Hilbert-Smith conjecture. To end with, we will explain how a certain special case of the conjecture can be interpreted as the non-contractibility of non-trivial compact quantum groups, and prove it for some classes of compact quantum groups. (Based on joint works with Paul F. Baum, Ludwik Dabrowski, Eusebio Gardella, Sergey Neshveyev, Mariusz Tobolski and Jianchao Wu.)